Asymptotics of dynamic lattice Green’s functions

Vanel, Alice L.; Colquitt, Daniel; Makwana, Mehul P.

doi:10.1016/j.wavemoti.2016.05.010

Cited by 13 publications

(11 citation statements)

References 29 publications

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“…Forced vibrations of grid of beams has been considered for a two-dimensional mass/spring periodic structure [23], while asymptotic approximations of lattice Green's functions have been given [24,25], close to standing wave frequencies, with the purpose of revealing the directional anisotropy in two and three-dimensional periodic lattices.…”

Section: Introductionmentioning

confidence: 99%

Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

Bordiga

Cabras

Bigoni

2019

International Journal of Solids and Structures

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In-plane wave propagation in a periodic rectangular grid beam structure, which includes rotational inertia (so-called 'Rayleigh beams'), is analyzed both with a Floquet-Bloch exact formulation for free oscillations and with a numerical treatment (developed with PML absorbing boundary conditions) for forced vibrations (including Fourier representation and energy flux evaluations), induced by a concentrated force or moment. A complex interplay is observed between axial and flexural vibrations (not found in the common idealization of out-of-plane motion), giving rise to several forms of vibration localization: 'X-', 'cross-' and 'star-' shaped, and channel propagation. These localizations are triggered by several factors, including rotational inertia and slenderness of the beams and the type of forcing source (concentrated force or moment). Although the considered grid of beams introduces an orthotropy in the mechanical response, a surprising 'isotropization' of the vibration is observed at special frequencies. Moreover, rotational inertia is shown to 'sharpen' degeneracies related to Dirac cones (which become more pronounced when the aspect ratio of the grid is increased), while the slenderness can be tuned to achieve a perfectly flat band in the dispersion diagram. The obtained results can be exploited in the realization of metamaterials designed to control wave propagation. and the solution techniques are well-known, many interesting features still remain to be explored. This exploration is provided in the present article, where an exact Floquet-Bloch analysis is performed and complemented with a numerical treatment of the forced vibrations induced by the application of a concentrated force or moment, including presentation of the Fourier transform and energy flow (treated in [26] for free vibrations). It is shown that (i.) aspect ratio of the grid, (ii.) slenderness and (iii.) rotational inertia of the beams decide the emergence of several forms of highly-localized waveforms, namely, 'channel propagation', 'X-', 'cross-', 'star-' shaped vibration modes. Moreover, these mechanical properties of the grid can be designed to obtain flat bands and degeneracies related to Dirac cones in the dispersion diagram and directional anisotropy or, surprisingly, dynamic 'isotropization', for which waves propagate in a square lattice with the polar symmetry characterizing propagation in an isotropic medium.The presented results open the way to the design of vibrating devices with engineered properties, to achieve control of elastic wave propagation.2 In-plane Floquet-Bloch waves in a rectangular grid of beams An infinite lattice of Rayleigh beams is considered, periodically arranged in a rectangular geometry as shown in Fig. 1a, together with the unit cell, Fig. 1b.

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Section: Introductionmentioning

confidence: 99%

Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

Bordiga

Cabras

Bigoni

2019

International Journal of Solids and Structures

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“…We draw upon [22] where the discrete effective medium is created, the coefficients at frequency f I have T 11 = T 22 = −8.6 and T 33 = 17.2, showing the effective medium to have indefinite medium behaviour at frequency f I . We note that in [22], such an effective medium is termed hyperbolic medium, but it differs from the physics described in [14]. To validate these model predictions, the experimental PC, as shown in the inset of Fig.…”

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confidence: 99%

“…2(a) that shows these surfaces overlain. Moreover, for the mass-spring system highly directive anisotropy occurs at critical points in the Brillouin zone and these are associated with effective anisotropic media [22], akin to indefinite media [14]. Given the striking similarity of the isofrequency surfaces vis-à-vis the discrete and continuous models we draw conclusions from the discrete model and transfer them to the continuum model thereby avoiding large numbers of expensive simulations.…”

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confidence: 99%

“…It is particularly notable that these squares intersect at 8 points, one of which we label I for future reference, and for which the direction ΓI points along ΓR to the corners of the Brillouin zone. This point I is where three lines cross, and for which the group velocity directed along those lines is zero, clearly the group velocity itself is not completely zero but this intersection creates a critical point and energy is preferentially directed along ΓI; the discrete theory is given by [22]. Since the full isofrequency surfaces are captured by the discrete model, see Fig.…”

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confidence: 99%

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Acoustic flat lensing using an indefinite medium

et al. 2019

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Acoustic flat lensing is achieved here by tuning a phononic array to have indefinite medium behaviour in a narrow frequency spectral region along the acoustic branch. This is confirmed by the occurrence of a flat band along an unusual path in the Brillouin zone and by interpreting the intersection point of isofrequency contours on the corresponding isofrequency surface; coherent directive beams are formed whose reflection from the array surfaces create lensing. Theoretical predictions are corroborated by time-domain experiments, airborne acoustic waves generated by a source with a frequency centered about 10.6 kHz, placed at three different distances from one side of a finite phononic crystal slab, constructed from polymeric spheres, yield distinctive focal spots on the other side. These experiments evaluate the pressure field using optical feedback interferometry and demonstrate precise control of the three-dimensional wave trajectory through a sonic crystal.The band spectra of photonic [1, 2] and phononic [3] crystals can be interpreted to predict a rich array of interesting physical effects, for instance anomalous refraction [1,4] and all-angle-negative refraction [5] amongst many others; understanding these spectra underpins advances in electronic properties, wave transport in photonics and acoustics, as well as in interference phenomena throughout many fields of physical and engineering sciences.In this Letter we report experimental results where an image of a volumetric source through a three-dimensional phononic crystal (PC) forms according to the physics of indefinite media [6], see Fig. 1. The image is not created by tilting the crystal as in acoustic superlenses [7], or at low frequencies using effective media [8] nor by negative refraction acoustic flat lenses using metamaterials [9,10]. Instead we identify critical points on the isofrequency surfaces, for a simple cubic array of rigid spheres, where beam-like trajectories are formed and use these beams, and their reflections, to create lensing; this is using the properties of indefinite media [6].The first experimental demonstration of a threedimensional flat acoustic lens in 2004 [7] used 0.8-mm tungsten carbide beads surrounded by water, with the beads closely packed in a face-centered cubic crystal structure along the body diagonal (ΓR crystal direction); the lensing function was above the phononic band gap of the PC and at 1.57 MHz with the pressure waves focused into a tight spot (about 5 mm). We give an alternative design to this well-known phononic lens, based upon a different physical mechanism, also exhibiting focusing reminiscent of the Veselago-Pendry convergent flat lens [11,12], see Fig. 1. Negative refraction and superlensing of underwater pressure waves has been theorized with an anisotropic acoustic metamaterial formed by layers of perforated rigid plates [13], that is analogous to a hyperbolic medium in electromagnetism [14].As in [7] we use an array of sound-hard spheres, although now in air, take a primitive cubic array, and take ad...

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“…In Fig. 3 we show illustrative simulations in the case of a quasi-2D slab of boxes, performed using the methodology from [32]. These examples demonstrate the strongly frequency-dependent nature of the fields; referring back to Fig.…”

Section: A Monopole Point Sourcementioning

confidence: 98%

Asymptotic Modeling of Phononic Box Crystals

Vanel¹,

Schnitzer²

2019

SIAM J. Appl. Math.

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We introduce phononic box crystals, namely arrays of adjoined perforated boxes, as a threedimensional prototype for an unusual class of subwavelength metamaterials based on directly coupling resonating elements. In this case, when the holes coupling the boxes are small, we create networks of Helmholtz resonators with nearest-neighbour interactions. We use matched asymptotic expansions, in the small hole limit, to derive simple, yet asymptotically accurate, discrete wave equations governing the pressure field. These network equations readily furnish analytical dispersion relations for box arrays, slabs and crystals, that agree favourably with finiteelement simulations of the physical problem. Our results reveal that the entire acoustic branch is uniformly squeezed into a subwavelength regime; consequently, phononic box crystals exhibit nonlinear-dispersion effects (such as dynamic anisotropy) in a relatively wide band, as well as a high effective refractive index in the long-wavelength limit. We also study the sound field produced by sources placed within one of the boxes by comparing and contrasting monopole-with dipole-type forcing; for the former the pressure field is asymptotically enhanced whilst for the latter there is no asymptotic enhancement and the translation from the microscale to the discrete description entails evaluating singular limits, using a regularized and efficient scheme, of the Neumann's Green's function for a cube. We conclude with an example of using our asymptotic framework to calculate localized modes trapped within a defected box array.

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Asymptotics of dynamic lattice Green’s functions

Cited by 13 publications

References 29 publications

Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

Acoustic flat lensing using an indefinite medium

Asymptotic Modeling of Phononic Box Crystals

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